Group #22625404 — subgroups

Subgroups — #22625404

Order 9344.

212 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17989609	C₂	1	1	yes	maximal, characteristic
2	#17989546	C₂	1	1	yes	maximal, characteristic
3	#17989444	C₂	1	4	yes	maximal, complemented
4	#17989142	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924036	—	2	1	no	—
6	C₄⋊#10596833	—	2	1	no	—
7	#14556648	—	2	1	no	—
8	#14556646	—	2	1	no	—
9	#14556646	C₂²	1	2	yes	—
10	#14556646	C₂²	1	2	yes	—
11	#14556591	C₄	1	1	yes	characteristic
12	#14556582	C₂²	1	2	yes	—
13	#14556571	—	2	1	no	—
14	#14556567	—	2	1	no	—
15	Q₈×Dic₇₃	—	2	1	no	—
16	#14556556	—	2	1	no	—
17	#14556549	C₄	1	2	yes	—
18	C₄²×D₇₃	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₇₃	C₄	1	1	yes	characteristic
20	#11924036	—	2	4	no	—
21	C₂×#10596833	—	2	2	no	—
22	C₂×#10596833	—	2	2	no	—
23	C₂×Dic₁₄₆	—	2	1	no	—
24	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
26	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
27	C₄⋊Dic₇₃	—	2	1	no	—
28	C₄⋊Dic₇₃	—	2	2	no	—
29	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
30	#11924007	—	2	2	no	—
31	#11924005	—	2	2	no	—
32	C₄×Dic₇₃	—	2	2	no	—
33	C₄×Dic₇₃	D₄	1	1	yes	characteristic
34	C₄×Dic₇₃	D₄	1	1	yes	characteristic
35	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
36	C₂×Q₈×C₇₃	—	2	1	no	—
37	C₄⋊C₄×C₇₃	—	2	1	no	—
38	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
39	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
41	Q₈×C₇₃	—	4	2	no	—
42	C₂×C₄×C₇₃	—	2	2	no	—
43	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×Dic₇₃	—	2	2	no	—
47	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
50	C₄×D₇₃	—	2	2	no	—
51	C₄×D₇₃	—	2	2	no	—
52	C₄×D₇₃	—	4	1	no	—
53	Dic₁₄₆	—	4	2	no	—
54	#10596833	—	2	4	no	—
55	#10596833	—	4	2	no	—
56	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×D₇₃	—	2	1	no	—
58	C₂×D₇₃	—	2	1	no	—
59	C₂×D₇₃	#101	1	2	yes	—
60	C₄×C₇₃	—	2	1	no	—
61	C₄×C₇₃	—	2	1	no	—
62	C₄×C₇₃	—	4	2	no	—
63	C₄×C₇₃	#104	1	2	yes	—
64	Dic₇₃	—	2	1	no	—
65	Dic₇₃	—	2	1	no	—
66	Dic₇₃	—	2	1	no	—
67	Dic₇₃	—	4	2	no	—
68	C₂×C₇₃	C₂×#101	1	1	yes	characteristic
69	C₂×C₇₃	C₂×#104	1	1	yes	characteristic
70	C₂×C₇₃	#421	1	1	yes	characteristic
71	D₇₃	—	2	2	no	—
72	#1531	—	73	1	no	maximal, Sylow
73	C₇₃	#1531	1	1	yes	characteristic, complemented, Sylow
74	#533	—	73	1	no	—
75	#432	—	73	1	no	—
76	C₂×#97	—	73	1	no	—
77	#330	—	73	4	no	—
78	C₂²×C₈	—	146	1	no	—
79	C₄⋊Q₈	—	73	1	no	—
80	C₄⋊Q₈	#64408	1	1	yes	characteristic
81	#125	—	73	2	no	—
82	C₂²⋊Q₈	—	146	1	no	—
83	C₂²⋊Q₈	—	146	1	no	—
84	C₄×Q₈	—	146	1	no	—
85	C₄×Q₈	—	146	1	no	—
86	#117	—	73	2	no	—
87	C₂×C₄²	—	73	1	no	—
88	C₄⋊C₈	—	146	1	no	—
89	C₂²⋊C₈	—	146	1	no	—
90	#97	—	73	2	no	—
91	#97	—	73	2	no	—
92	#97	—	146	1	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	146	1	no	—
95	C₂²×C₄	—	73	1	no	—
96	C₂²×C₄	—	73	1	no	—
97	C₂²×C₄	—	73	1	no	—
98	C₂×C₈	—	146	2	no	—
99	C₂×C₈	—	146	2	no	—
100	C₂×C₈	—	146	4	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	73	2	no	—
103	C₄⋊C₄	—	146	2	no	—
104	C₄⋊C₄	—	146	2	no	—
105	C₄⋊C₄	—	146	1	no	—
106	C₄⋊C₄	C₂×#64408	1	2	yes	—
107	C₂²⋊C₄	—	146	2	no	—
108	C₄²	—	73	1	no	—
109	C₄²	—	73	1	no	—
110	C₄²	—	73	1	no	—
111	C₄²	—	146	2	no	—
112	C₄²	C₂×#64408	1	1	yes	characteristic
113	C₂³	—	73	1	no	—
114	Q₈	—	4	2	no	—
115	Q₈	—	292	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	292	1	no	—
118	C₂×C₄	—	73	1	no	—
119	C₂×C₄	—	73	1	no	—
120	C₂×C₄	—	73	1	no	—
121	C₂×C₄	—	146	2	no	—
122	C₂×C₄	—	146	2	no	—
123	C₂×C₄	—	146	2	no	—
124	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
125	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
126	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
127	C₈	—	292	2	no	—
128	C₈	—	146	4	no	—
129	C₂²	—	73	2	no	—
130	C₂²	—	146	1	no	—
131	C₂²	—	146	1	no	—
132	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	2	no	—
136	C₄	—	292	2	no	—
137	C₄	—	146	1	no	—
138	C₄	—	146	1	no	—
139	C₄	—	146	1	no	—
140	C₄	#14556655	1	2	yes	—
141	C₂	—	146	2	no	—
142	C₂	D₇₃⋊#101	1	1	yes	characteristic
143	C₂	C₂×#14556655	1	1	yes	characteristic
144	C₂	#17989586	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989609

C₂

yes

maximal, characteristic

#17989546

C₂

yes

maximal, characteristic

#17989444

C₂

yes

maximal, complemented

#17989142

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

#14556646

C₂²

yes

—

#14556591

C₄

yes

characteristic

#14556582

C₂²

yes

—

#14556571

—

#14556567

—

Q₈×Dic₇₃

—

#14556556

—

#14556549

C₄

yes

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₄

yes

—

—

—

C₂×Dic₁₄₆

—

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

#11924007

—

#11924005

—

C₄×Dic₇₃

—

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

C₂³

yes

characteristic

C₂×Q₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

C₂²⋊C₄

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

#10596833

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#101

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#104

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1531

—

maximal, Sylow

C₇₃

#1531

yes

characteristic, complemented, Sylow

#533

—

#432

—

C₂×#97

—

#330

—

C₂²×C₈

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

#64408

yes

characteristic

#125

—

C₂²⋊Q₈

—

146

—

C₂²⋊Q₈

—

146

—

C₄×Q₈

—

146

—

C₄×Q₈

—

146

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

146

—

#97

—

#97

—

#97

—

146

—

C₂×Q₈

—

C₂×Q₈

—

146

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

100

C₂×C₈

—

146

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

146

—

104

C₄⋊C₄

—

146

—

105

C₄⋊C₄

—

146

—

106

C₄⋊C₄

C₂×#64408

yes

—

107

C₂²⋊C₄

—

146

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

146

—

112

C₄²

C₂×#64408

yes

characteristic

113

C₂³

—

114

Q₈

—

115

Q₈

—

292

—

116

C₂×C₄

—

117

C₂×C₄

—

292

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

146

—

122

C₂×C₄

—

146

—

123

C₂×C₄

—

146

—

124

C₂×C₄

C₂²⋊#64408

yes

characteristic

125

C₂×C₄

C₂²⋊#64408

yes

characteristic

126

C₂×C₄

C₂²×#64408

yes

characteristic

127

C₈

—

292

—

128

C₈

—

146

—

129

C₂²

—

130

C₂²

—

146

—

131

C₂²

—

146

—

132

C₂²

C₂×C₂²⋊#64408

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

292

—

137

C₄

—

146

—

138

C₄

—

146

—

139

C₄

—

146

—

140

C₄

#14556655

yes

—

141

C₂

—

146

—

142

C₂

D₇₃⋊#101

yes

characteristic

143

C₂

C₂×#14556655

yes

characteristic

144

C₂

#17989586

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₄⋊Q₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	C₄⋊Q₈	#64408
hypocenter	C₇₃	#1531
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₄⋊Q₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

C₄⋊Q₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
3. #17989444	C₂	semidirect
73. C₇₃	#1531	semidirect

H ≅ G/N

type

3. #17989444

C₂

semidirect

73. C₇₃

#1531

semidirect

Subgroups — #22625404

Conjugacy classes of subgroups

Complements